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===Homeomorphism v. Diffeomorphism===

===Homeomorphism v. Diffeomorphism===

For those readers who have some experience with topology, there may be a question of whether a diffeomorphism always exists.  While the existence of a homeomorphism between manifolds demonstrates the existence of a diffeomorphism between them if they are of dimension <math>\leq 3 \ </math>, the h-cobordism theorem actually demonstrates that a diffeomorphism exists for <math>n\geq 5 \ </math>. The only open case is the four dimensional one.

For those readers who have some experience with topology, there may be a question of whether a diffeomorphism always exists.  While the existence of a homeomorphism between manifolds demonstrates the existence of a diffeomorphism between them if they are of dimension <math>\leq 3 \ </math>, the h-cobordism theorem actually demonstrates that a diffeomorphism exists for <math>n=5 \ </math>, but for <math>n\geq 6 \ </math> it is possible to construct objects which are homeomorphic and not diffeomorphic - such objects are called "exotic <math>\mathbb{S}^n</math>." The only open case is the four dimensional one.

==Proof of the Conjecture==

==Proof of the Conjecture==